 PositiveRoots Details - Maple Help

Details for PositiveRoots Description

 • For each root type "A", "B", "C", "D" there are general formulas for calculating a list of positive roots as linear combinations of a given list of simple roots. These linear combinations are given in several texts. See, for example Varadarajan Lie Groups, Lie Algebras and Their Representations, Section 4.4 or Cap and Slovak Parabolic Geometries I. Background and General Theory, Section 2.2.6. Here are these formulas as they are coded into the 3rd calling sequence for the command.
 •

part1 := seq(seq(([0$i, 1$j, 0$(ell - i - j)], j = 1 .. ell - i)), i = 0 .. ell + 1)];  • ${B}_{\mathrm{𝓁}}$ part1 := seq(seq(([0$i, 1$j, 0$(ell - i - j)], j = 1 .. ell - i)), i = 0 .. ell),

part2 := seq(seq([0$i, 1$(j), 2$(ell - i - j) ], j = 1 .. ell - i - 1), i = 0 .. ell);  • ${C}_{\mathrm{𝓁}}$ part1 := seq(seq(([0$i, 1$j, 0$(ell - i - j)], j = 1 .. ell - i)), i = 0 .. ell);

part2 := seq(seq([0$i, 1$(j), 2$(ell - i - j - 1) ,1], j = 1 .. ell - i - 2), i = 0 .. ell); part3 := seq([0$(i - 1), 2$(ell - i), 1], i = 1 .. ell-1);  • ${\mathrm{D}}_{\mathrm{𝓁}}$ part1 := seq(seq(([0$i, 1$j, 0$(ell - i - j)], j = 1 .. ell - i - 1)), i = 0 .. ell);

part2 := seq([0$(i - 1), 1$(ell - i - 1), 0, 1], i = 1 .. ell - 1);

part3 := seq(seq([0$i, 1$(j + 1), 2\$(ell - i - j -3) , 1, 1], j = 0 .. ell - i -3), i = 0 .. ell - 2);

 • For any Cartan matrix there is also a simple algorithm for calculating the positive roots from the Cartan matrix. This algorithm is presented in W.A. de Graaf, Lie Algebras: Theory and Algorithms, page 162 and also in W. Fulton and J. Harris, Representation Theory, A First Course, page 330. This approach is coded into the 4th calling sequence for the PositiveRoots command. Examples

 > $\mathrm{with}\left(\mathrm{DifferentialGeometry}\right):$$\mathrm{with}\left(\mathrm{LieAlgebras}\right):$

Example 1. Type A

 > $\mathrm{PositiveRoots}\left("A",2\right)$
 $\left[\left[\begin{array}{r}{1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {1}\end{array}\right]\right]$ (2.1)
 > $\mathrm{PositiveRoots}\left("A",3\right)$
 $\left[\left[\begin{array}{r}{1}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {0}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {1}\\ {1}\end{array}\right]\right]$ (2.2)
 > $\mathrm{PositiveRoots}\left("A",4\right)$
 $\left[\left[\begin{array}{r}{1}\\ {0}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {0}\\ {1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {0}\\ {0}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {1}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {0}\\ {1}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {1}\\ {1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {1}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {1}\\ {1}\\ {1}\end{array}\right]\right]$ (2.3)
 > $\mathrm{PositiveRoots}\left("A",4\right)$
 $\left[\left[\begin{array}{r}{1}\\ {0}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {0}\\ {1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {0}\\ {0}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {1}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {0}\\ {1}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {1}\\ {1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {1}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {1}\\ {1}\\ {1}\end{array}\right]\right]$ (2.4)
 > $\mathrm{CM}≔\mathrm{CartanMatrix}\left("A",4\right)$
 ${\mathrm{CM}}{:=}\left[\begin{array}{rrrr}{2}& {-}{1}& {0}& {0}\\ {-}{1}& {2}& {-}{1}& {0}\\ {0}& {-}{1}& {2}& {-}{1}\\ {0}& {0}& {-}{1}& {2}\end{array}\right]$ (2.5)
 > $\mathrm{PositiveRoots}\left(\mathrm{CM}\right)$
 $\left[\left[\begin{array}{r}{1}\\ {0}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {0}\\ {1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {0}\\ {0}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {1}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {0}\\ {1}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {1}\\ {1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {1}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {1}\\ {1}\\ {1}\end{array}\right]\right]$ (2.6)

Example 2. Type B

 > $\mathrm{PositiveRoots}\left("B",2\right)$
 $\left[\left[\begin{array}{r}{1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {2}\end{array}\right]\right]$ (2.7)
 > $\mathrm{PositiveRoots}\left("B",3\right)$
 $\left[\left[\begin{array}{r}{1}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {0}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {1}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {2}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {1}\\ {2}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {2}\\ {2}\end{array}\right]\right]$ (2.8)
 > $\mathrm{PositiveRoots}\left("B",4\right)$
 $\left[\left[\begin{array}{r}{1}\\ {0}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {0}\\ {1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {0}\\ {0}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {1}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {0}\\ {1}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {1}\\ {1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {1}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {0}\\ {1}\\ {2}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {1}\\ {1}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {1}\\ {2}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {1}\\ {1}\\ {2}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {2}\\ {2}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {1}\\ {2}\\ {2}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {2}\\ {2}\\ {2}\end{array}\right]\right]$ (2.9)
 > $\mathrm{CM}≔\mathrm{CartanMatrix}\left("B",4\right)$
 ${\mathrm{CM}}{:=}\left[\begin{array}{rrrr}{2}& {-}{1}& {0}& {0}\\ {-}{1}& {2}& {-}{1}& {0}\\ {0}& {-}{1}& {2}& {-}{2}\\ {0}& {0}& {-}{1}& {2}\end{array}\right]$ (2.10)
 > $\mathrm{PositiveRoots}\left(\mathrm{CM}\right)$
 $\left[\left[\begin{array}{r}{1}\\ {0}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {0}\\ {1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {0}\\ {0}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {1}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {0}\\ {1}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {1}\\ {1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {1}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {0}\\ {1}\\ {2}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {1}\\ {1}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {1}\\ {2}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {1}\\ {1}\\ {2}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {2}\\ {2}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {1}\\ {2}\\ {2}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {2}\\ {2}\\ {2}\end{array}\right]\right]$ (2.11)

Example 3. Type C

Note that is the same as apart from an ordering of the components.

 > $\mathrm{PositiveRoots}\left("C",2\right)$
 $\left[\left[\begin{array}{r}{1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{2}\\ {1}\end{array}\right]\right]$ (2.12)
 > $\mathrm{PositiveRoots}\left("C",3\right)$
 $\left[\left[\begin{array}{r}{1}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {0}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {1}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {2}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {2}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{2}\\ {2}\\ {1}\end{array}\right]\right]$ (2.13)
 > $\mathrm{PositiveRoots}\left("C",4\right)$
 $\left[\left[\begin{array}{r}{1}\\ {0}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {0}\\ {1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {0}\\ {0}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {1}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {0}\\ {1}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {1}\\ {1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {1}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {0}\\ {2}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {1}\\ {1}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {2}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {1}\\ {2}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {2}\\ {2}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {2}\\ {2}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{2}\\ {2}\\ {2}\\ {1}\end{array}\right]\right]$ (2.14)
 > $\mathrm{CM}≔\mathrm{CartanMatrix}\left("C",4\right)$
 ${\mathrm{CM}}{:=}\left[\begin{array}{rrrr}{2}& {-}{1}& {0}& {0}\\ {-}{1}& {2}& {-}{1}& {0}\\ {0}& {-}{1}& {2}& {-}{1}\\ {0}& {0}& {-}{2}& {2}\end{array}\right]$ (2.15)
 > $\mathrm{PositiveRoots}\left(\mathrm{CM}\right)$
 $\left[\left[\begin{array}{r}{1}\\ {0}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {0}\\ {1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {0}\\ {0}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {1}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {0}\\ {1}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {1}\\ {1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {1}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {0}\\ {2}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {1}\\ {1}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {2}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {1}\\ {2}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {2}\\ {2}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {2}\\ {2}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{2}\\ {2}\\ {2}\\ {1}\end{array}\right]\right]$ (2.16)

Example 4. Type D

Note that and ${A}_{3}$coincide.

 > $\mathrm{PositiveRoots}\left("D",2\right)$
 $\left[\left[\begin{array}{r}{1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\end{array}\right]\right]$ (2.17)
 > $\mathrm{PositiveRoots}\left("D",3\right)$
 $\left[\left[\begin{array}{r}{1}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {0}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {0}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {1}\\ {1}\end{array}\right]\right]$ (2.18)
 > $\mathrm{PositiveRoots}\left("D",4\right)$
 $\left[\left[\begin{array}{r}{1}\\ {0}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {0}\\ {1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {0}\\ {0}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {1}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {0}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {1}\\ {1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {1}\\ {0}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {1}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {1}\\ {1}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {2}\\ {1}\\ {1}\end{array}\right]\right]$ (2.19)
 > $\mathrm{CM}≔\mathrm{CartanMatrix}\left("D",4\right)$
 ${\mathrm{CM}}{:=}\left[\begin{array}{rrrr}{2}& {-}{1}& {0}& {0}\\ {-}{1}& {2}& {-}{1}& {-}{1}\\ {0}& {-}{1}& {2}& {0}\\ {0}& {-}{1}& {0}& {2}\end{array}\right]$ (2.20)
 > $\mathrm{PositiveRoots}\left(\mathrm{CM}\right)$
 $\left[\left[\begin{array}{r}{1}\\ {0}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {0}\\ {1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {0}\\ {0}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {1}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {0}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {1}\\ {1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {1}\\ {0}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {1}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {1}\\ {1}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {2}\\ {1}\\ {1}\end{array}\right]\right]$ (2.21)